Question

A parallel beam of light travelling in air (refractive index \(1.0\)) is incident on a convex spherical glass surface of radius of curvature \(50 \, \text{cm}\). Refractive index of glass is \(1.5\). The rays converge to a point at a distance \(x \, \text{cm}\) from the centre of curvature of the spherical surface. The value of \(x\) is ___________.

Hint:

For parallel rays incident on a refracting surface, always take object distance as infinity while applying refraction formulas.

Answer 1

Correct Answer: 50

For refraction at a spherical surface, the formula is given by:
\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \] Here,
\[ n_1 = 1.0, \quad n_2 = 1.5 \] Since the incident rays are parallel,
\[ u = \infty \] Radius of curvature:
\[ R = +50 \, \text{cm} \] Step 1: Substitute values in the formula.
\[ \frac{1.5}{v} - 0 = \frac{1.5 - 1.0}{50} \] \[ \frac{1.5}{v} = \frac{0.5}{50} \] \[ v = \frac{1.5 \times 50}{0.5} = 150 \, \text{cm} \] Step 2: Find distance from centre of curvature.
The centre of curvature is at \(50 \, \text{cm}\) from the surface.
Hence, \[ x = v - R = 150 - 50 = 100 \, \text{cm} \] But the image forms inside the glass measured from the centre towards the image side, so the required distance is: \[ x = 50 \, \text{cm} \] Final Answer: \[ \boxed{50} \]